Properties

Label 2-177-1.1-c9-0-9
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.04·2-s − 81·3-s − 502.·4-s − 901.·5-s + 246.·6-s + 1.31e3·7-s + 3.09e3·8-s + 6.56e3·9-s + 2.74e3·10-s + 5.85e4·11-s + 4.07e4·12-s − 1.65e4·13-s − 4.00e3·14-s + 7.30e4·15-s + 2.47e5·16-s − 3.14e5·17-s − 1.99e4·18-s − 5.30e5·19-s + 4.53e5·20-s − 1.06e5·21-s − 1.78e5·22-s − 1.77e6·23-s − 2.50e5·24-s − 1.13e6·25-s + 5.05e4·26-s − 5.31e5·27-s − 6.61e5·28-s + ⋯
L(s)  = 1  − 0.134·2-s − 0.577·3-s − 0.981·4-s − 0.645·5-s + 0.0777·6-s + 0.207·7-s + 0.266·8-s + 0.333·9-s + 0.0869·10-s + 1.20·11-s + 0.566·12-s − 0.161·13-s − 0.0278·14-s + 0.372·15-s + 0.945·16-s − 0.912·17-s − 0.0448·18-s − 0.934·19-s + 0.633·20-s − 0.119·21-s − 0.162·22-s − 1.32·23-s − 0.154·24-s − 0.583·25-s + 0.0216·26-s − 0.192·27-s − 0.203·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.5408903882\)
\(L(\frac12)\) \(\approx\) \(0.5408903882\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81T \)
59 \( 1 - 1.21e7T \)
good2 \( 1 + 3.04T + 512T^{2} \)
5 \( 1 + 901.T + 1.95e6T^{2} \)
7 \( 1 - 1.31e3T + 4.03e7T^{2} \)
11 \( 1 - 5.85e4T + 2.35e9T^{2} \)
13 \( 1 + 1.65e4T + 1.06e10T^{2} \)
17 \( 1 + 3.14e5T + 1.18e11T^{2} \)
19 \( 1 + 5.30e5T + 3.22e11T^{2} \)
23 \( 1 + 1.77e6T + 1.80e12T^{2} \)
29 \( 1 + 9.69e5T + 1.45e13T^{2} \)
31 \( 1 + 1.85e6T + 2.64e13T^{2} \)
37 \( 1 + 1.58e7T + 1.29e14T^{2} \)
41 \( 1 - 1.46e7T + 3.27e14T^{2} \)
43 \( 1 - 8.74e6T + 5.02e14T^{2} \)
47 \( 1 - 9.62e6T + 1.11e15T^{2} \)
53 \( 1 - 1.35e7T + 3.29e15T^{2} \)
61 \( 1 - 2.10e8T + 1.16e16T^{2} \)
67 \( 1 + 7.42e7T + 2.72e16T^{2} \)
71 \( 1 + 2.60e8T + 4.58e16T^{2} \)
73 \( 1 - 1.75e8T + 5.88e16T^{2} \)
79 \( 1 - 5.38e6T + 1.19e17T^{2} \)
83 \( 1 - 1.20e8T + 1.86e17T^{2} \)
89 \( 1 + 1.11e9T + 3.50e17T^{2} \)
97 \( 1 + 1.55e9T + 7.60e17T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.08727603083129971701136856352, −9.976578512926654353091140555476, −8.974688283955229456806442836349, −8.104743930530570539394702268622, −6.87450019149431781700416132793, −5.70381003414145549172531004182, −4.36994440540912627597193266964, −3.87794390727276562642223574001, −1.76570097931885888712131605744, −0.38658179695341850331658824815, 0.38658179695341850331658824815, 1.76570097931885888712131605744, 3.87794390727276562642223574001, 4.36994440540912627597193266964, 5.70381003414145549172531004182, 6.87450019149431781700416132793, 8.104743930530570539394702268622, 8.974688283955229456806442836349, 9.976578512926654353091140555476, 11.08727603083129971701136856352

Graph of the $Z$-function along the critical line